The 28-year-old bagged his first goal since September on 34 minutes when he headed home Kelyn Rowe's cross, shortly after Bobby Shuttleworth struck the bar for the hosts, who were looking to extend their seven-game winning streak on home turf.
Chris Tierney curled home a free kick from 20 yards out to double New England's advantage, although Justin Meram converted from inside the box 10 minutes later to pull Columbus right back into the game.
That two-goal cushion was soon regained, however, when Lee Nguyen burst forward and found the back of the net at the end of his run to make it 3-1.
New England, who finished three points ahead of their opponents in the regular MLS season, looked to have put the two-legged tie beyond doubt when Davies curled home his second 12 minutes before time.
But Federico Higuain converted from the spot in the dying seconds to give the Crew some real hope ahead of the return game at the Gillette Stadium next week.